\(\int \frac {\log (c (d+e x^n)^p)}{x^2} \, dx\) [74]

Optimal result

Integrand size = 16, antiderivative size = 66 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x^2} \, dx=-\frac {e n p x^{-1+n} \operatorname {Hypergeometric2F1}\left (1,-\frac {1-n}{n},2-\frac {1}{n},-\frac {e x^n}{d}\right )}{d (1-n)}-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{x} \]

[Out]

-e*n*p*x^(-1+n)*hypergeom([1, (-1+n)/n],[2-1/n],-e*x^n/d)/d/(1-n)-ln(c*(d+e*x^n)^p)/x

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2505, 371} \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x^2} \, dx=-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{x}-\frac {e n p x^{n-1} \operatorname {Hypergeometric2F1}\left (1,-\frac {1-n}{n},2-\frac {1}{n},-\frac {e x^n}{d}\right )}{d (1-n)} \]

[In]

Int[Log[c*(d + e*x^n)^p]/x^2,x]

[Out]

-((e*n*p*x^(-1 + n)*Hypergeometric2F1[1, -((1 - n)/n), 2 - n^(-1), -((e*x^n)/d)])/(d*(1 - n))) - Log[c*(d + e*
x^n)^p]/x

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 2505

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[(f*x)^(m +
1)*((a + b*Log[c*(d + e*x^n)^p])/(f*(m + 1))), x] - Dist[b*e*n*(p/(f*(m + 1))), Int[x^(n - 1)*((f*x)^(m + 1)/
(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {\log \left (c \left (d+e x^n\right )^p\right )}{x}+(e n p) \int \frac {x^{-2+n}}{d+e x^n} \, dx \\ & = -\frac {e n p x^{-1+n} \, _2F_1\left (1,-\frac {1-n}{n};2-\frac {1}{n};-\frac {e x^n}{d}\right )}{d (1-n)}-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.89 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x^2} \, dx=\frac {\frac {e n p x^n \operatorname {Hypergeometric2F1}\left (1,\frac {-1+n}{n},2-\frac {1}{n},-\frac {e x^n}{d}\right )}{d (-1+n)}-\log \left (c \left (d+e x^n\right )^p\right )}{x} \]

[In]

Integrate[Log[c*(d + e*x^n)^p]/x^2,x]

[Out]

((e*n*p*x^n*Hypergeometric2F1[1, (-1 + n)/n, 2 - n^(-1), -((e*x^n)/d)])/(d*(-1 + n)) - Log[c*(d + e*x^n)^p])/x

Maple [F]

\[\int \frac {\ln \left (c \left (d +e \,x^{n}\right )^{p}\right )}{x^{2}}d x\]

[In]

int(ln(c*(d+e*x^n)^p)/x^2,x)

[Out]

int(ln(c*(d+e*x^n)^p)/x^2,x)

Fricas [F]

\[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x^2} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x^{2}} \,d x } \]

[In]

integrate(log(c*(d+e*x^n)^p)/x^2,x, algorithm="fricas")

[Out]

integral(log((e*x^n + d)^p*c)/x^2, x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.46 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.11 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x^2} \, dx=- \frac {\log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{x} + \frac {d^{\frac {1}{n}} d^{1 - \frac {1}{n}} e e^{- \frac {1}{n}} e^{-1 + \frac {1}{n}} p \Phi \left (\frac {d x^{- n} e^{i \pi }}{e}, 1, \frac {1}{n}\right ) \Gamma \left (- \frac {1}{n}\right )}{d n x \Gamma \left (1 - \frac {1}{n}\right )} \]

[In]

integrate(ln(c*(d+e*x**n)**p)/x**2,x)

[Out]

-log(c*(d + e*x**n)**p)/x + d**(1/n)*d**(1 - 1/n)*e*e**(-1 + 1/n)*p*lerchphi(d*exp_polar(I*pi)/(e*x**n), 1, 1/
n)*gamma(-1/n)/(d*e**(1/n)*n*x*gamma(1 - 1/n))

Maxima [F]

\[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x^2} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x^{2}} \,d x } \]

[In]

integrate(log(c*(d+e*x^n)^p)/x^2,x, algorithm="maxima")

[Out]

-d*n*p*integrate(1/(e*x^2*x^n + d*x^2), x) - (n*p + log((e*x^n + d)^p) + log(c))/x

Giac [F]

\[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x^2} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x^{2}} \,d x } \]

[In]

integrate(log(c*(d+e*x^n)^p)/x^2,x, algorithm="giac")

[Out]

integrate(log((e*x^n + d)^p*c)/x^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x^2} \, dx=\int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}{x^2} \,d x \]

[In]

int(log(c*(d + e*x^n)^p)/x^2,x)

[Out]

int(log(c*(d + e*x^n)^p)/x^2, x)